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The Arabic number system links number length with magnitude: longer numbers represent larger values. In numerical comparisons, interference arises when physical and numerical dimensions are misaligned, and processing improves with greater numerical distance. While ERP studies have dissociated numerical processing effects, evidence for early encoding of numerical information has been limited due to confounding visual properties. In two ERP experiments, we examined whether number length influences early multi-digit number processing, independent of visual size, by using scribbled line patterns to control physical length. Experiment 1 ( N  = 27) compared digit values in tie numbers (numbers with repeated digits) to standard “555”, ignoring number length. Experiment 2 ( N  = 27) judged number length, ignoring digit identity. Targets varied in number length and numerical distance, creating congruent and incongruent conditions. ERP results revealed enhanced parieto-occipital N1 negativity (~ 120–150 ms) for longer numbers, indicating early number length encoding independent of overall visual size. Central P2p (~ 150–190 ms) reflected refined numerical distance processing. Centro-parietal P3 (~ 300–360 ms) showed congruity effects only when numerical magnitude was task-relevant. These findings reveal three distinct processing stages: early encoding (N1), refined magnitude comparison (P2p), and conflict resolution (P3). This novel neurophysiological evidence showcases the unique influence of number length and numerical syntax on magnitude evaluation.

Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for \"different\" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1–9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers, specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.

Post-Traumatic Stress Disorder (PTSD) research indicates that hyper-reactivity to trauma-related stimuli reflects reduced prefrontal cortex (PFC) modulation of amygdala reactivity. However, other studies indicate a dissociative “shutdown” reaction to overwhelming aversive stimuli, possibly reflecting PFC over-modulation. To explore this, we used an Event-Related Potential (ERP) oddball paradigm to study P3 responses in the presence of the following: 1. Trauma-unrelated morbid distractors (e.g., “injured bear”) related to the Rorschach inkblot test, and 2. Negative distractors (e.g., “significant failure”), among participants with high post-traumatic stress symptoms (PTS; n = 20), low PTS (n = 17), and controls (n = 15). Distractors were presented at 20% frequency amongst the more frequent (60%) neutral standard stimuli (e.g., “desk lamp”) and the equally frequent (20%) neutral trauma-unrelated target stimulus (“golden fish”). P3 amplitudes were high in the presence of morbid distractors and low in the presence of negative distractors only amongst the control group. Possible mechanisms underlying the lack of P3 amplitude modulation after trauma are discussed.

Past research indicates that concepts of infinity are not fully understood. In countably infinite sets, infinity is presumed to be perceived as larger than any finite natural number. This study explored whether symbolic representations of infinity are processed as such through contrasts with Arabic and verbal written numbers. Comparisons between the infinity word and number words were responded to faster than comparisons of two number words, but not when the infinity symbol was solely compared to Arabic numbers. Moreover, infinity comparisons yielded distance-like effects, suggesting that infinity (both word and symbol) can be misconceived as a “natural number” closer to larger numbers than small ones. These findings demonstrate difficulty perceiving the physically smallest stimulus (∞) as the upper end-value and seem to reflect a limited understanding of symbolic forms of infinity among adults. They further highlight the impact of notation and numerical syntax on how we process symbolic numerical information.

Exponential expressions represent series that grow at a fast pace such as carbon pollution and the spread of disease. Despite their importance, people tend to struggle with these expressions. In two experiments, participants chose the larger of two exponential expressions as quickly and accurately as possible. We manipulated the distance between the base/power components and their compatibility. In base-power compatible pairs, both the base and power of one expression were larger than the other (e.g., 2 3 vs. 3 4 ), while in base-power incompatible pairs, the base of one expression was larger than the base in the other expression but the relation between the power components of the two expressions was reversed (e.g., 3 2 vs. 2 4 ). Moreover, while in the first experiment the larger power always led to the larger result, in the second experiment we introduced base-result congruent pairs as well. Namely, the larger base led to the larger result. Our results showed a base-power compatibility effect, which was also larger for larger power distances (Experiments 1 – 2 ). Furthermore, participants processed the base-result congruent pairs faster and more accurately than the power-result congruent pairs (Experiment 2 ). These findings suggest that while both the base and power components are processed when comparing exponential expressions, the base is more salient. This exemplifies an incorrect processing of the syntax of exponential expressions, where the power typically has a larger mathematical contribution to the result of the expression.

We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set – the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

Previous research has shown that multi-digit number processing is modulated by both place-value and physical size of the digits. By pitting place-value against physical size, the present study examined whether one of the attributes had a greater impact on the automatic processing of multi-digit numbers. In three experiments, participants were presented with two-digit number pairs that appeared in frames. They were instructed to select the larger frame while ignoring the numbers within the frames. Importantly, we manipulated the physical size of the digits (i.e., both decade/unit digits were physically larger) within the frames, the unit–decade compatibility (i.e., the relationship between the numerical values of both decade and unit digits was consistent or inconsistent), and the congruity between the numerical values of the decade digits and the frames’ physical size (i.e., decade-value–frame-size congruity). In Experiment 1 , where all pairs were unit–decade compatible, a decade-value–frame-size congruity effect emerged for pairs with physically larger decade, but not unit, digits. However, when adding unit–decade incompatible pairs (Experiments 2 – 3 ), in unit–decade compatible pairs, there was a decade-value–frame-size congruity effect regardless of the digits’ physical size. In contrast, in unit–decade incompatible pairs, there was no decade-value–frame-size congruity effect, even when the physically larger digit (i.e., unit) contradicted the place-value information, presumably due to the cancellation of the opposing influences of the digits’ physical sizes their place-values. Overall, these findings suggest that place-value and physical size are intertwined in the Hindu–Arabic numerical system and are processed as one.

Research in cognitive science has documented numerous phenomena that are approximated by linear relationships. In the domain of numerical cognition, the use of linear regression for estimating linear effects (e.g., distance and SNARC effects) became common following Fias, Brysbaert, Geypens, and d’Ydewalle’s ( 1996 ) study on the SNARC effect. While their work has become the model for analyzing linear effects in the field, it requires statistical analysis of individual participants and does not provide measures of the proportions of variability accounted for (cf. Lorch & Myers, 1990 ). In the present methodological note, using both the distance and SNARC effects as examples, we demonstrate how linear effects can be estimated in a simple way within the framework of repeated measures analysis of variance. This method allows for estimating effect sizes in terms of both slope and proportions of variability accounted for. Finally, we show that our method can easily be extended to estimate linear interaction effects, not just linear effects calculated as main effects.

Number processing evokes spatial biases, both when dealing with single digits and in more complex mental calculations. Here we investigated whether these two biases have a common origin, by examining their flexibility. Participants pointed to the locations of arithmetic results on a visually presented line with an inverted, right-to-left number arrangement. We found directionally opposite spatial biases for mental arithmetic and for a parity task administered both before and after the arithmetic task. We discuss implications of this dissociation in our results for the task-dependent cognitive representation of numbers.